Faulhaber's formula

Faulhaber's formula is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

In mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers as a (p + 1)th-degree polynomial function of n, the coefficients involving Bernoulli numbers.

Generate the first 10 closed-form expressions, starting with p = 0.

C

Translation of: Modula-2
#include <stdbool.h>#include <stdio.h>#include <stdlib.h> int binomial(int n, int k) {    int num, denom, i;     if (n < 0 || k < 0 || n < k) return -1;    if (n == 0 || k == 0) return 1;     num = 1;    for (i = k + 1; i <= n; ++i) {        num = num * i;    }     denom = 1;    for (i = 2; i <= n - k; ++i) {        denom *= i;    }     return num / denom;} int gcd(int a, int b) {    int temp;    while (b != 0) {        temp = a % b;        a = b;        b = temp;    }    return a;} typedef struct tFrac {    int num, denom;} Frac; Frac makeFrac(int n, int d) {    Frac result;    int g;     if (d == 0) {        result.num = 0;        result.denom = 0;        return result;    }     if (n == 0) {        d = 1;    } else if (d < 0) {        n = -n;        d = -d;    }     g = abs(gcd(n, d));    if (g > 1) {        n = n / g;        d = d / g;    }     result.num = n;    result.denom = d;    return result;} Frac negateFrac(Frac f) {    return makeFrac(-f.num, f.denom);} Frac subFrac(Frac lhs, Frac rhs) {    return makeFrac(lhs.num * rhs.denom - lhs.denom * rhs.num, rhs.denom * lhs.denom);} Frac multFrac(Frac lhs, Frac rhs) {    return makeFrac(lhs.num * rhs.num, lhs.denom * rhs.denom);} bool equalFrac(Frac lhs, Frac rhs) {    return (lhs.num == rhs.num) && (lhs.denom == rhs.denom);} bool lessFrac(Frac lhs, Frac rhs) {    return (lhs.num * rhs.denom) < (rhs.num * lhs.denom);} void printFrac(Frac f) {    printf("%d", f.num);    if (f.denom != 1) {        printf("/%d", f.denom);    }} Frac bernoulli(int n) {    Frac a[16];    int j, m;     if (n < 0) {        a[0].num = 0;        a[0].denom = 0;        return a[0];    }     for (m = 0; m <= n; ++m) {        a[m] = makeFrac(1, m + 1);        for (j = m; j >= 1; --j) {            a[j - 1] = multFrac(subFrac(a[j - 1], a[j]), makeFrac(j, 1));        }    }     if (n != 1) {        return a[0];    }     return negateFrac(a[0]);} void faulhaber(int p) {    Frac coeff, q;    int j, pwr, sign;     printf("%d : ", p);    q = makeFrac(1, p + 1);    sign = -1;    for (j = 0; j <= p; ++j) {        sign = -1 * sign;        coeff = multFrac(multFrac(multFrac(q, makeFrac(sign, 1)), makeFrac(binomial(p + 1, j), 1)), bernoulli(j));        if (equalFrac(coeff, makeFrac(0, 1))) {            continue;        }        if (j == 0) {            if (!equalFrac(coeff, makeFrac(1, 1))) {                if (equalFrac(coeff, makeFrac(-1, 1))) {                    printf("-");                } else {                    printFrac(coeff);                }            }        } else {            if (equalFrac(coeff, makeFrac(1, 1))) {                printf(" + ");            } else if (equalFrac(coeff, makeFrac(-1, 1))) {                printf(" - ");            } else if (lessFrac(makeFrac(0, 1), coeff)) {                printf(" + ");                printFrac(coeff);            } else {                printf(" - ");                printFrac(negateFrac(coeff));            }        }        pwr = p + 1 - j;        if (pwr > 1) {            printf("n^%d", pwr);        } else {            printf("n");        }    }    printf("\n");} int main() {    int i;     for (i = 0; i < 10; ++i) {        faulhaber(i);    }     return 0;}
Output:
0 : n
1 : 1/2n^2 + 1/2n
2 : 1/3n^3 + 1/2n^2 + 1/6n
3 : 1/4n^4 + 1/2n^3 + 1/4n^2
4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n
5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2
6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n
7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2
8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n
9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2

C++

Translation of: D

Uses C++17

#include <iostream>#include <numeric>#include <sstream>#include <vector> class Frac {public:	Frac(long n, long d) {		if (d == 0) {			throw new std::runtime_error("d must not be zero");		} 		long nn = n;		long dd = d;		if (nn == 0) {			dd = 1;		} else if (dd < 0) {			nn = -nn;			dd = -dd;		} 		long g = abs(std::gcd(nn, dd));		if (g > 1) {			nn /= g;			dd /= g;		} 		num = nn;		denom = dd;	} 	Frac operator-() const {		return Frac(-num, denom);	} 	Frac operator+(const Frac& rhs) const {		return Frac(num*rhs.denom + denom * rhs.num, rhs.denom*denom);	} 	Frac operator-(const Frac& rhs) const {		return Frac(num*rhs.denom - denom * rhs.num, rhs.denom*denom);	} 	Frac operator*(const Frac& rhs) const {		return Frac(num*rhs.num, denom*rhs.denom);	} 	bool operator==(const Frac& rhs) const {		return num == rhs.num && denom == rhs.denom;	} 	bool operator!=(const Frac& rhs) const {		return num != rhs.num || denom != rhs.denom;	} 	bool operator<(const Frac& rhs) const {		if (denom == rhs.denom) {			return num < rhs.num;		}		return num * rhs.denom < rhs.num * denom;	} 	friend std::ostream& operator<<(std::ostream&, const Frac&); 	static Frac ZERO() {		return Frac(0, 1);	} 	static Frac ONE() {		return Frac(1, 1);	} private:	long num;	long denom;}; std::ostream & operator<<(std::ostream & os, const Frac &f) {	if (f.num == 0 || f.denom == 1) {		return os << f.num;	} 	std::stringstream ss;	ss << f.num << "/" << f.denom;	return os << ss.str();} Frac bernoulli(int n) {	if (n < 0) {		throw new std::runtime_error("n may not be negative or zero");	} 	std::vector<Frac> a;	for (int m = 0; m <= n; m++) {		a.push_back(Frac(1, m + 1));		for (int j = m; j >= 1; j--) {			a[j - 1] = (a[j - 1] - a[j]) * Frac(j, 1);		}	} 	// returns 'first' Bernoulli number	if (n != 1) return a[0];	return -a[0];} int binomial(int n, int k) {	if (n < 0 || k < 0 || n < k) {		throw new std::runtime_error("parameters are invalid");	}	if (n == 0 || k == 0) return 1; 	int num = 1;	for (int i = k + 1; i <= n; i++) {		num *= i;	} 	int denom = 1;	for (int i = 2; i <= n - k; i++) {		denom *= i;	} 	return num / denom;} void faulhaber(int p) {	using namespace std;	cout << p << " : "; 	auto q = Frac(1, p + 1);	int sign = -1;	for (int j = 0; j < p + 1; j++) {		sign *= -1;		auto coeff = q * Frac(sign, 1) * Frac(binomial(p + 1, j), 1) * bernoulli(j);		if (coeff == Frac::ZERO()) {			continue;		}		if (j == 0) {			if (coeff == -Frac::ONE()) {				cout << "-";			} else if (coeff != Frac::ONE()) {				cout << coeff;			}		} else {			if (coeff == Frac::ONE()) {				cout << " + ";			} else if (coeff == -Frac::ONE()) {				cout << " - ";			} else if (coeff < Frac::ZERO()) {				cout << " - " << -coeff;			} else {				cout << " + " << coeff;			}		}		int pwr = p + 1 - j;		if (pwr > 1) {			cout << "n^" << pwr;		} else {			cout << "n";		}	}	cout << endl;} int main() {	for (int i = 0; i < 10; i++) {		faulhaber(i);	} 	return 0;}
Output:
0 : n
1 : 1/2n^2 + 1/2n
2 : 1/3n^3 + 1/2n^2 + 1/6n
3 : 1/4n^4 + 1/2n^3 + 1/4n^2
4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n
5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2
6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n
7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2
8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n
9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2

C#

Translation of: Java
using System; namespace FaulhabersFormula {    internal class Frac {        private long num;        private long denom;         public static readonly Frac ZERO = new Frac(0, 1);        public static readonly Frac ONE = new Frac(1, 1);         public Frac(long n, long d) {            if (d == 0) {                throw new ArgumentException("d must not be zero");            }            long nn = n;            long dd = d;            if (nn == 0) {                dd = 1;            }            else if (dd < 0) {                nn = -nn;                dd = -dd;            }            long g = Math.Abs(Gcd(nn, dd));            if (g > 1) {                nn /= g;                dd /= g;            }            num = nn;            denom = dd;        }         private static long Gcd(long a, long b) {            if (b == 0) {                return a;            }            return Gcd(b, a % b);        }         public static Frac operator -(Frac self) {            return new Frac(-self.num, self.denom);        }         public static Frac operator +(Frac lhs, Frac rhs) {            return new Frac(lhs.num * rhs.denom + lhs.denom * rhs.num, rhs.denom * lhs.denom);        }         public static Frac operator -(Frac lhs, Frac rhs) {            return lhs + -rhs;        }         public static Frac operator *(Frac lhs, Frac rhs) {            return new Frac(lhs.num * rhs.num, lhs.denom * rhs.denom);        }         public static bool operator <(Frac lhs, Frac rhs) {            double x = (double)lhs.num / lhs.denom;            double y = (double)rhs.num / rhs.denom;            return x < y;        }         public static bool operator >(Frac lhs, Frac rhs) {            double x = (double)lhs.num / lhs.denom;            double y = (double)rhs.num / rhs.denom;            return x > y;        }         public static bool operator ==(Frac lhs, Frac rhs) {            return lhs.num == rhs.num && lhs.denom == rhs.denom;        }         public static bool operator !=(Frac lhs, Frac rhs) {            return lhs.num != rhs.num || lhs.denom != rhs.denom;        }         public override string ToString() {            if (denom == 1) {                return num.ToString();            }            return string.Format("{0}/{1}", num, denom);        }         public override bool Equals(object obj) {            var frac = obj as Frac;            return frac != null &&                   num == frac.num &&                   denom == frac.denom;        }         public override int GetHashCode() {            var hashCode = 1317992671;            hashCode = hashCode * -1521134295 + num.GetHashCode();            hashCode = hashCode * -1521134295 + denom.GetHashCode();            return hashCode;        }    }     class Program {        static Frac Bernoulli(int n) {            if (n < 0) {                throw new ArgumentException("n may not be negative or zero");            }            Frac[] a = new Frac[n + 1];            for (int m = 0; m <= n; m++) {                a[m] = new Frac(1, m + 1);                for (int j = m; j >= 1; j--) {                    a[j - 1] = (a[j - 1] - a[j]) * new Frac(j, 1);                }            }            // returns 'first' Bernoulli number            if (n != 1) return a[0];            return -a[0];        }         static int Binomial(int n, int k) {            if (n < 0 || k < 0 || n < k) {                throw new ArgumentException();            }            if (n == 0 || k == 0) return 1;            int num = 1;            for (int i = k + 1; i <= n; i++) {                num = num * i;            }            int denom = 1;            for (int i = 2; i <= n - k; i++) {                denom = denom * i;            }            return num / denom;        }         static void Faulhaber(int p) {            Console.Write("{0} : ", p);            Frac q = new Frac(1, p + 1);            int sign = -1;            for (int j = 0; j <= p; j++) {                sign *= -1;                Frac coeff = q * new Frac(sign, 1) * new Frac(Binomial(p + 1, j), 1) * Bernoulli(j);                if (Frac.ZERO == coeff) continue;                if (j == 0) {                    if (Frac.ONE != coeff) {                        if (-Frac.ONE == coeff) {                            Console.Write("-");                        }                        else {                            Console.Write(coeff);                        }                    }                }                else {                    if (Frac.ONE == coeff) {                        Console.Write(" + ");                    }                    else if (-Frac.ONE == coeff) {                        Console.Write(" - ");                    }                    else if (Frac.ZERO < coeff) {                        Console.Write(" + {0}", coeff);                    }                    else {                        Console.Write(" - {0}", -coeff);                    }                }                int pwr = p + 1 - j;                if (pwr > 1) {                    Console.Write("n^{0}", pwr);                }                else {                    Console.Write("n");                }            }            Console.WriteLine();        }         static void Main(string[] args) {            for (int i = 0; i < 10; i++) {                Faulhaber(i);            }        }    }}
Output:
0 : n
1 : 1/2n^2 + 1/2n
2 : 1/3n^3 + 1/2n^2 + 1/6n
3 : 1/4n^4 + 1/2n^3 + 1/4n^2
4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n
5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2
6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n
7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2
8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n
9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2

D

Translation of: Kotlin
import std.algorithm : fold;import std.exception : enforce;import std.format : formattedWrite;import std.numeric : cmp, gcd;import std.range : iota;import std.stdio;import std.traits; auto abs(T)(T val)if (isNumeric!T) {    if (val < 0) {        return -val;    }    return val;} struct Frac {    long num;    long denom;     enum ZERO = Frac(0, 1);    enum ONE = Frac(1, 1);     this(long n, long d) in {        enforce(d != 0, "Parameter d may not be zero.");    } body {        auto nn = n;        auto dd = d;        if (nn == 0) {            dd = 1;        } else if (dd < 0) {            nn = -nn;            dd = -dd;        }        auto g = gcd(abs(nn), abs(dd));        if (g > 1) {            nn /= g;            dd /= g;        }        num = nn;        denom = dd;    }     auto opBinary(string op)(Frac rhs) const {        static if (op == "+" || op == "-") {            return mixin("Frac(num*rhs.denom"~op~"denom*rhs.num, rhs.denom*denom)");        } else if (op == "*") {            return Frac(num*rhs.num, denom*rhs.denom);        }    }     auto opUnary(string op : "-")() const {        return Frac(-num, denom);    }     int opCmp(Frac rhs) const {        return cmp(cast(real) this, cast(real) rhs);    }     bool opEquals(Frac rhs) const {        return num == rhs.num && denom == rhs.denom;    }     void toString(scope void delegate(const(char)[]) sink) const {        if (denom == 1) {            formattedWrite(sink, "%d", num);        } else {            formattedWrite(sink, "%d/%s", num, denom);        }    }     T opCast(T)() const if (isFloatingPoint!T) {        return cast(T) num / denom;    }} auto abs(Frac f) {    if (f.num >= 0) {        return f;    }    return -f;} auto bernoulli(int n) in {    enforce(n >= 0, "Parameter n must not be negative.");} body {    Frac[] a;    a.length = n+1;    a[0] = Frac.ZERO;    foreach (m; 0..n+1) {        a[m] = Frac(1, m+1);        foreach_reverse (j; 1..m+1) {            a[j-1] = (a[j-1] - a[j]) * Frac(j, 1);        }    }    if (n != 1) {        return a[0];    }    return -a[0];} auto binomial(int n, int k) in {    enforce(n>=0 && k>=0 && n>=k);} body {    if (n==0 || k==0) return 1;    auto num = iota(k+1, n+1).fold!"a*b"(1);    auto den = iota(2, n-k+1).fold!"a*b"(1);    return num / den;} auto faulhaber(int p) {    write(p, " : ");    auto q = Frac(1, p+1);    auto sign = -1;    foreach (j; 0..p+1) {        sign *= -1;        auto coeff = q * Frac(sign, 1) * Frac(binomial(p+1, j), 1) * bernoulli(j);        if (coeff == Frac.ZERO) continue;        if (j == 0) {            if (coeff == -Frac.ONE) {                write("-");            } else if (coeff != Frac.ONE) {                write(coeff);            }        } else {            if (coeff == Frac.ONE) {                write(" + ");            } else if (coeff == -Frac.ONE) {                write(" - ");            } else if (coeff > Frac.ZERO) {                write(" + ", coeff);            } else {                write(" - ", -coeff);            }        }        auto pwr = p + 1 - j;        if (pwr > 1) {            write("n^", pwr);        } else {            write("n");        }    }    writeln;} void main() {    foreach (i; 0..10) {        faulhaber(i);    }}
Output:
0 : n
1 : 1/2n^2 + 1/2n
2 : 1/3n^3 + 1/2n^2 + 1/6n
3 : 1/4n^4 + 1/2n^3 + 1/4n^2
4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n
5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2
6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n
7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2
8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n
9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2

EchoLisp

 (lib 'math) ;; for bernoulli numbers(string-delimiter "") ;; returns list of polynomial coefficients(define (Faulhaber p)	(cons 0	(for/list ([k (in-range p -1 -1)])		(* (Cnp (1+ p) k) (bernoulli k))))) ;; prints formal polynomial(define (task (pmax 10))    (for ((p pmax))     (writeln p 'โ  (/ 1 (1+ p)) '* (poly->string 'n (Faulhaber p))))) ;; extra credit - compute sums(define (Faulcomp n p)	(printf "ฮฃ(1..%d) n^%d = %d" n p (/  (poly n (Faulhaber p)) (1+ p) )))
Output:
(task)
0     โ     1     *     n
1     โ     1/2     *     n^2 + n
2     โ     1/3     *     n^3 + 3/2 n^2 + 1/2 n
3     โ     1/4     *     n^4 + 2 n^3 + n^2
4     โ     1/5     *     n^5 + 5/2 n^4 + 5/3 n^3 -1/6 n
5     โ     1/6     *     n^6 + 3 n^5 + 5/2 n^4 -1/2 n^2
6     โ     1/7     *     n^7 + 7/2 n^6 + 7/2 n^5 -7/6 n^3 + 1/6 n
7     โ     1/8     *     n^8 + 4 n^7 + 14/3 n^6 -7/3 n^4 + 2/3 n^2
8     โ     1/9     *     n^9 + 9/2 n^8 + 6 n^7 -21/5 n^5 + 2 n^3 -3/10 n
9     โ     1/10     *     n^10 + 5 n^9 + 15/2 n^8 -7 n^6 + 5 n^4 -3/2 n^2

(Faulcomp 100 2)
ฮฃ(1..100) n^2 = 338350
(Faulcomp 100 1)
ฮฃ(1..100) n^1 = 5050

(lib 'bigint)
(Faulcomp 100 9)
ฮฃ(1..100) n^9 = 10507499300049998000

;; check it ...
(for/sum ((n 101)) (expt n 9))
โ 10507499300049998500


GAP

Straightforward implementation using GAP polynomials, and two different formulas: one based on Stirling numbers of the second kind (sum1, see Python implementation below in this page), and the usual Faulhaber formula (sum2). No optimization is made (one could compute Stirling numbers row by row, or the product in sum1 may be kept from one call to the other). Notice the Bernoulli term in the first formula is here only to correct the value of sum1(0), which is off by one because sum1 computes sums from 0 to n.

n := X(Rationals, "n");sum1 := p -> Sum([0 .. p], k -> Stirling2(p, k) * Product([0 .. k], j -> n + 1 - j) / (k + 1)) + 2 * Bernoulli(2 * p + 1);sum2 := p -> Sum([0 .. p], j -> (-1)^j * Binomial(p + 1, j) * Bernoulli(j) * n^(p + 1 - j)) / (p + 1);ForAll([0 .. 20], k -> sum1(k) = sum2(k)); for p in [0 .. 9] do    Print(sum1(p), "\n");od; n1/2*n^2+1/2*n1/3*n^3+1/2*n^2+1/6*n1/4*n^4+1/2*n^3+1/4*n^21/5*n^5+1/2*n^4+1/3*n^3-1/30*n1/6*n^6+1/2*n^5+5/12*n^4-1/12*n^21/7*n^7+1/2*n^6+1/2*n^5-1/6*n^3+1/42*n1/8*n^8+1/2*n^7+7/12*n^6-7/24*n^4+1/12*n^21/9*n^9+1/2*n^8+2/3*n^7-7/15*n^5+2/9*n^3-1/30*n1/10*n^10+1/2*n^9+3/4*n^8-7/10*n^6+1/2*n^4-3/20*n^2

Go

package main import (	"fmt"	"math/big") func bernoulli(z *big.Rat, n int64) *big.Rat {	if z == nil {		z = new(big.Rat)	}	a := make([]big.Rat, n+1)	for m := range a {		a[m].SetFrac64(1, int64(m+1))		for j := m; j >= 1; j-- {			d := &a[j-1]			d.Mul(z.SetInt64(int64(j)), d.Sub(d, &a[j]))		}	}	return z.Set(&a[0])} func main() {	// allocate needed big.Rat's once	q := new(big.Rat)	c := new(big.Rat)      // coefficients	be := new(big.Rat)     // for Bernoulli numbers	bi := big.NewRat(1, 1) // for binomials 	for p := int64(0); p < 10; p++ {		fmt.Print(p, " : ")		q.SetFrac64(1, p+1)		neg := true		for j := int64(0); j <= p; j++ {			neg = !neg			if neg {				c.Neg(q)			} else {				c.Set(q)			}			bi.Num().Binomial(p+1, j)			bernoulli(be, j)			c.Mul(c, bi)			c.Mul(c, be)			if c.Num().BitLen() == 0 {				continue			}			if j == 0 {				fmt.Printf(" %4s", c.RatString())			} else {				fmt.Printf(" %+2d/%-2d", c.Num(), c.Denom())			}			fmt.Print("รn")			if exp := p + 1 - j; exp > 1 {				fmt.Printf("^%-2d", exp)			}		}		fmt.Println()	}}
Output:
0 :     1รn
1 :   1/2รn^2  -1/2 รn
2 :   1/3รn^3  -1/2 รn^2  +1/6 รn
3 :   1/4รn^4  -1/2 รn^3  +1/4 รn^2
4 :   1/5รn^5  -1/2 รn^4  +1/3 รn^3  -1/30รn
5 :   1/6รn^6  -1/2 รn^5  +5/12รn^4  -1/12รn^2
6 :   1/7รn^7  -1/2 รn^6  +1/2 รn^5  -1/6 รn^3  +1/42รn
7 :   1/8รn^8  -1/2 รn^7  +7/12รn^6  -7/24รn^4  +1/12รn^2
8 :   1/9รn^9  -1/2 รn^8  +2/3 รn^7  -7/15รn^5  +2/9 รn^3  -1/30รn
9 :  1/10รn^10 -1/2 รn^9  +3/4 รn^8  -7/10รn^6  +1/2 รn^4  -3/20รn^2


import Data.Ratio ((%), numerator, denominator)import Data.List (intercalate, transpose)import Control.Arrow ((&&&), (***))import Data.Char (isSpace)import Data.Monoid ((<>)) -- FAULHABER -------------------------------------------------------------------faulhaber :: [[Rational]]faulhaber =  tail $scanl (\rs n -> let xs = zipWith ((*) . (n %)) [2 ..] rs in 1 - sum xs : xs) [] [0 ..] -- EXPRESSION STRINGS ----------------------------------------------------------polynomials :: [[(String, String)]]polynomials = fmap ((ratioPower =<<) . reverse . flip zip [1 ..]) faulhaber -- Rows of (Power string, Ratio string) tuples -> Printable linesexpressionTable :: [[(String, String)]] -> [String]expressionTable ps = let cols = transpose (fullTable ps) in expressionRow <$>     zip       [0 ..]       (transpose \$        zipWith          ($$lw, rw) col -> (fmap (justifyLeft lw ' ' *** justifyLeft rw ' ') col)) (colWidths cols) cols) -- Value pair -> String pair (lifted into list for use with >>=)ratioPower :: (Rational, Integer) -> [(String, String)]ratioPower (nd, j) = let (num, den) = (numerator &&& denominator) nd sn | num == 0 = [] | (j /= 1) = ("n^" <> show j) | otherwise = "n" sr | num == 0 = [] | den == 1 && num == 1 = [] | den == 1 = show num <> "n" | otherwise = intercalate "/" [show num, show den] s = sr <> sn in if null s then [] else [(sn, sr)] -- Rows of uneven length -> All rows padded to length of longestfullTable :: [[(String, String)]] -> [[(String, String)]]fullTable xs = let lng = maximum  length <> xs in (<>) <*> (flip replicate ([], []) . (-) lng . length) <> xs justifyLeft :: Int -> Char -> String -> StringjustifyLeft n c s = take n (s <> replicate n c) -- (Row index, Expression pairs) -> String joined by conjunctionsexpressionRow :: (Int, [(String, String)]) -> StringexpressionRow (i, row) = concat [ show i , " -> " , foldr (\s a -> concat [ s , if blank a || head a == '-' then " " else " + " , a ]) "" (polyTerm <> row) ] -- (Power string, Ratio String) -> Combined string with possible '*'polyTerm :: (String, String) -> StringpolyTerm (l, r) | blank l || blank r = l <> r | head r == '-' = concat ["- ", l, " * ", tail r] | otherwise = intercalate " * " [l, r] blank :: String -> Boolblank = all isSpace -- Maximum widths of power and ratio elements in each columncolWidths :: [[(String, String)]] -> [(Int, Int)]colWidths = fmap (foldr (\(ls, rs) (lMax, rMax) -> (max (length ls) lMax, max (length rs) rMax)) (0, 0)) -- Length of string excluding any leading '-'unsignedLength :: String -> IntunsignedLength xs = let l = length xs in case l of 0 -> 0 _ -> case head xs of '-' -> l - 1 _ -> l -- TEST ------------------------------------------------------------------------main :: IO ()main = (putStrLn . unlines . expressionTable . take 10) polynomials Output: 0 -> n 1 -> n^2 * 1/2 + n * 1/2 2 -> n^3 * 1/3 + n^2 * 1/2 + n * 1/6 3 -> n^4 * 1/4 + n^3 * 1/2 + n^2 * 1/4 4 -> n^5 * 1/5 + n^4 * 1/2 + n^3 * 1/3 - n * 1/30 5 -> n^6 * 1/6 + n^5 * 1/2 + n^4 * 5/12 - n^2 * 1/12 6 -> n^7 * 1/7 + n^6 * 1/2 + n^5 * 1/2 - n^3 * 1/6 + n * 1/42 7 -> n^8 * 1/8 + n^7 * 1/2 + n^6 * 7/12 - n^4 * 7/24 + n^2 * 1/12 8 -> n^9 * 1/9 + n^8 * 1/2 + n^7 * 2/3 - n^5 * 7/15 + n^3 * 2/9 - n * 1/30 9 -> n^10 * 1/10 + n^9 * 1/2 + n^8 * 3/4 - n^6 * 7/10 + n^4 * 1/2 - n^2 * 3/20 J Implementation: Bsecond=:verb define"0 +/,(<:*(_1^[)*!*(y^~1+[)%1+])"0/~i.1x+y) Bfirst=: Bsecond - 1&= Faul=:adverb define (0,|.(%m+1x) * (_1x&^ * !&(m+1) * Bfirst) i.1+m)&p.) Task example:  0 Faul0 1x&p. 1 Faul0 1r2 1r2&p. 2 Faul0 1r6 1r2 1r3&p. 3 Faul0 0 1r4 1r2 1r4&p. 4 Faul0 _1r30 0 1r3 1r2 1r5&p. 5 Faul0 0 _1r12 0 5r12 1r2 1r6&p. 6 Faul0 1r42 0 _1r6 0 1r2 1r2 1r7&p. 7 Faul0 0 1r12 0 _7r24 0 7r12 1r2 1r8&p. 8 Faul0 _1r30 0 2r9 0 _7r15 0 2r3 1r2 1r9&p. 9 Faul0 0 _3r20 0 1r2 0 _7r10 0 3r4 1r2 1r10&p. Double checking our work:  Fcheck=: dyad def'+/(1+i.y)^x'"0 9 Faul i.50 1 513 20196 282340 9 Fcheck i.50 1 513 20196 282340 2 Faul i.50 1 5 14 30 2 Fcheck i.50 1 5 14 30 Java Translation of: Kotlin Works with: Java version 8 import java.util.Arrays;import java.util.stream.IntStream; public class FaulhabersFormula { private static long gcd(long a, long b) { if (b == 0) { return a; } return gcd(b, a % b); } private static class Frac implements Comparable<Frac> { private long num; private long denom; public static final Frac ZERO = new Frac(0, 1); public static final Frac ONE = new Frac(1, 1); public Frac(long n, long d) { if (d == 0) throw new IllegalArgumentException("d must not be zero"); long nn = n; long dd = d; if (nn == 0) { dd = 1; } else if (dd < 0) { nn = -nn; dd = -dd; } long g = Math.abs(gcd(nn, dd)); if (g > 1) { nn /= g; dd /= g; } num = nn; denom = dd; } public Frac plus(Frac rhs) { return new Frac(num * rhs.denom + denom * rhs.num, rhs.denom * denom); } public Frac unaryMinus() { return new Frac(-num, denom); } public Frac minus(Frac rhs) { return this.plus(rhs.unaryMinus()); } public Frac times(Frac rhs) { return new Frac(this.num * rhs.num, this.denom * rhs.denom); } @Override public int compareTo(Frac o) { double diff = toDouble() - o.toDouble(); return Double.compare(diff, 0.0); } @Override public boolean equals(Object obj) { return null != obj && obj instanceof Frac && this.compareTo((Frac) obj) == 0; } @Override public String toString() { if (denom == 1) { return Long.toString(num); } return String.format("%d/%d", num, denom); } private double toDouble() { return (double) num / denom; } } private static Frac bernoulli(int n) { if (n < 0) throw new IllegalArgumentException("n may not be negative or zero"); Frac[] a = new Frac[n + 1]; Arrays.fill(a, Frac.ZERO); for (int m = 0; m <= n; ++m) { a[m] = new Frac(1, m + 1); for (int j = m; j >= 1; --j) { a[j - 1] = a[j - 1].minus(a[j]).times(new Frac(j, 1)); } } // returns 'first' Bernoulli number if (n != 1) return a[0]; return a[0].unaryMinus(); } private static int binomial(int n, int k) { if (n < 0 || k < 0 || n < k) throw new IllegalArgumentException(); if (n == 0 || k == 0) return 1; int num = IntStream.rangeClosed(k + 1, n).reduce(1, (a, b) -> a * b); int den = IntStream.rangeClosed(2, n - k).reduce(1, (acc, i) -> acc * i); return num / den; } private static void faulhaber(int p) { System.out.printf("%d : ", p); Frac q = new Frac(1, p + 1); int sign = -1; for (int j = 0; j <= p; ++j) { sign *= -1; Frac coeff = q.times(new Frac(sign, 1)).times(new Frac(binomial(p + 1, j), 1)).times(bernoulli(j)); if (Frac.ZERO.equals(coeff)) continue; if (j == 0) { if (!Frac.ONE.equals(coeff)) { if (Frac.ONE.unaryMinus().equals(coeff)) { System.out.print("-"); } else { System.out.print(coeff); } } } else { if (Frac.ONE.equals(coeff)) { System.out.print(" + "); } else if (Frac.ONE.unaryMinus().equals(coeff)) { System.out.print(" - "); } else if (coeff.compareTo(Frac.ZERO) > 0) { System.out.printf(" + %s", coeff); } else { System.out.printf(" - %s", coeff.unaryMinus()); } } int pwr = p + 1 - j; if (pwr > 1) System.out.printf("n^%d", pwr); else System.out.print("n"); } System.out.println(); } public static void main(String[] args) { for (int i = 0; i <= 9; ++i) { faulhaber(i); } }} Output: 0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2 Julia Translation of: Kotlin Module: module Faulhaber function bernoulli(n::Integer) n โฅ 0 || throw(DomainError(n, "n must be a positive-or-0 number")) a = fill(0 // 1, n + 1) for m in 1:n a[m] = 1 // (m + 1) for j in m:-1:2 a[j - 1] = (a[j - 1] - a[j]) * j end end return ifelse(n != 1, a[1], -a[1])end const _exponents = collect(Char, "โฐยนยฒยณโดโตโถโทโธโน")toexponent(n) = join(_exponents[reverse(digits(n)) .+ 1]) function formula(p::Integer) print(p, ": ") q = 1 // (p + 1) s = -1 for j in 0:p s *= -1 coeff = q * s * binomial(p + 1, j) * bernoulli(j) iszero(coeff) && continue if iszero(j) print(coeff == 1 ? "" : coeff == -1 ? "-" : "coeff") else print(coeff == 1 ? " + " : coeff == -1 ? " - " : coeff > 0 ? " + coeff " : " - (-coeff) ") end pwr = p + 1 - j if pwr > 1 print("n", toexponent(pwr)) else print("n") end end println()end end # module Faulhaber Main: Faulhaber.formula.(1:10) Output: 1: + 1//2 n 2: + 1//2 nยฒ + 1//3 n 3: + 1//2 nยณ + 1//2 nยฒ - 1//6 n 4: + 1//2 nโด + 2//3 nยณ - 1//3 nยฒ + 1//30 n 5: + 1//2 nโต + 5//6 nโด - 5//9 nยณ + 1//12 nยฒ + 1//30 n 6: + 1//2 nโถ + nโต - 5//6 nโด + 1//6 nยณ + 1//10 nยฒ - 1//42 n 7: + 1//2 nโท + 7//6 nโถ - 7//6 nโต + 7//24 nโด + 7//30 nยณ - 1//12 nยฒ - 1//42 n 8: + 1//2 nโธ + 4//3 nโท - 14//9 nโถ + 7//15 nโต + 7//15 nโด - 2//9 nยณ - 2//21 nยฒ + 1//30 n 9: + 1//2 nโน + 3//2 nโธ - 2//1 nโท + 7//10 nโถ + 21//25 nโต - 1//2 nโด - 2//7 nยณ + 3//20 nยฒ + 1//30 n 10: + 1//2 nยนโฐ + 5//3 nโน - 5//2 nโธ + nโท + 7//5 nโถ - nโต - 5//7 nโด + 1//2 nยณ + 1//6 nยฒ - 5//66 n Kotlin As Kotlin doesn't have support for rational numbers built in, a cut-down version of the Frac class in the Arithmetic/Rational task has been used in order to express the polynomial coefficients as fractions. // version 1.1.2 fun gcd(a: Long, b: Long): Long = if (b == 0L) a else gcd(b, a % b) class Frac : Comparable<Frac> { val num: Long val denom: Long companion object { val ZERO = Frac(0, 1) val ONE = Frac(1, 1) } constructor(n: Long, d: Long) { require(d != 0L) var nn = n var dd = d if (nn == 0L) { dd = 1 } else if (dd < 0) { nn = -nn dd = -dd } val g = Math.abs(gcd(nn, dd)) if (g > 1) { nn /= g dd /= g } num = nn denom = dd } constructor(n: Int, d: Int) : this(n.toLong(), d.toLong()) operator fun plus(other: Frac) = Frac(num * other.denom + denom * other.num, other.denom * denom) operator fun unaryMinus() = Frac(-num, denom) operator fun minus(other: Frac) = this + (-other) operator fun times(other: Frac) = Frac(this.num * other.num, this.denom * other.denom) fun abs() = if (num >= 0) this else -this override fun compareTo(other: Frac): Int { val diff = this.toDouble() - other.toDouble() return when { diff < 0.0 -> -1 diff > 0.0 -> +1 else -> 0 } } override fun equals(other: Any?): Boolean { if (other == null || other !is Frac) return false return this.compareTo(other) == 0 } override fun toString() = if (denom == 1L) "num" else "num/denom" fun toDouble() = num.toDouble() / denom} fun bernoulli(n: Int): Frac { require(n >= 0) val a = Array<Frac>(n + 1) { Frac.ZERO } for (m in 0..n) { a[m] = Frac(1, m + 1) for (j in m downTo 1) a[j - 1] = (a[j - 1] - a[j]) * Frac(j, 1) } return if (n != 1) a[0] else -a[0] // returns 'first' Bernoulli number} fun binomial(n: Int, k: Int): Int { require(n >= 0 && k >= 0 && n >= k) if (n == 0 || k == 0) return 1 val num = (k + 1..n).fold(1) { acc, i -> acc * i } val den = (2..n - k).fold(1) { acc, i -> acc * i } return num / den} fun faulhaber(p: Int) { print("p : ") val q = Frac(1, p + 1) var sign = -1 for (j in 0..p) { sign *= -1 val coeff = q * Frac(sign, 1) * Frac(binomial(p + 1, j), 1) * bernoulli(j) if (coeff == Frac.ZERO) continue if (j == 0) { print(when { coeff == Frac.ONE -> "" coeff == -Frac.ONE -> "-" else -> "coeff" }) } else { print(when { coeff == Frac.ONE -> " + " coeff == -Frac.ONE -> " - " coeff > Frac.ZERO -> " + coeff" else -> " - {-coeff}" }) } val pwr = p + 1 - j if (pwr > 1) print("n^{p + 1 - j}") else print("n") } println()} fun main(args: Array<String>) { for (i in 0..9) faulhaber(i)} Output: 0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2  Lua Translation of: C function binomial(n,k) if n<0 or k<0 or n<k then return -1 end if n==0 or k==0 then return 1 end local num = 1 for i=k+1,n do num = num * i end local denom = 1 for i=2,n-k do denom = denom * i end return num / denomend function gcd(a,b) while b ~= 0 do local temp = a % b a = b b = temp end return aend function makeFrac(n,d) local result = {} if d==0 then result.num = 0 result.denom = 0 return result end if n==0 then d = 1 elseif d < 0 then n = -n d = -d end local g = math.abs(gcd(n, d)) if g>1 then n = n / g d = d / g end result.num = n result.denom = d return resultend function negateFrac(f) return makeFrac(-f.num, f.denom)end function subFrac(lhs, rhs) return makeFrac(lhs.num * rhs.denom - lhs.denom * rhs.num, rhs.denom * lhs.denom)end function multFrac(lhs, rhs) return makeFrac(lhs.num * rhs.num, lhs.denom * rhs.denom)end function equalFrac(lhs, rhs) return (lhs.num == rhs.num) and (lhs.denom == rhs.denom)end function lessFrac(lhs, rhs) return (lhs.num * rhs.denom) < (rhs.num * lhs.denom)end function printFrac(f) io.write(f.num) if f.denom ~= 1 then io.write("/"..f.denom) end return nilend function bernoulli(n) if n<0 then return {num=0, denom=0} end local a = {} for m=0,n do a[m] = makeFrac(1, m+1) for j=m,1,-1 do a[j-1] = multFrac(subFrac(a[j-1], a[j]), makeFrac(j, 1)) end end if n~=1 then return a[0] end return negateFrac(a[0])end function faulhaber(p) io.write(p.." : ") local q = makeFrac(1, p+1) local sign = -1 for j=0,p do sign = -1 * sign local coeff = multFrac(multFrac(multFrac(q, makeFrac(sign, 1)), makeFrac(binomial(p + 1, j), 1)), bernoulli(j)) if not equalFrac(coeff, makeFrac(0, 1)) then if j==0 then if not equalFrac(coeff, makeFrac(1, 1)) then if equalFrac(coeff, makeFrac(-1, 1)) then io.write("-") else printFrac(coeff) end end else if equalFrac(coeff, makeFrac(1, 1)) then io.write(" + ") elseif equalFrac(coeff, makeFrac(-1, 1)) then io.write(" - ") elseif lessFrac(makeFrac(0, 1), coeff) then io.write(" + ") printFrac(coeff) else io.write(" - ") printFrac(negateFrac(coeff)) end end local pwr = p + 1 - j if pwr>1 then io.write("n^"..pwr) else io.write("n") end end end print() return nilend -- mainfor i=0,9 do faulhaber(i)end Output: 0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2 Maxima sum1(p):=sum(stirling2(p,k)*pochhammer(n-k+1,k+1)/(k+1),k,0,p)sum2(p):=sum((-1)^j*binomial(p+1,j)*bern(j)*n^(p-j+1),j,0,p)/(p+1) makelist(expand(sum1(p)-sum2(p)),p,1,10);[0,0,0,0,0,0,0,0,0,0] for p from 0 thru 9 do print(expand(sum2(p))); Output: n n^2/2+n/2 n^3/3+n^2/2+n/6 n^4/4+n^3/2+n^2/4 n^5/5+n^4/2+n^3/3-n/30 n^6/6+n^5/2+(5*n^4)/12-n^2/12 n^7/7+n^6/2+n^5/2-n^3/6+n/42 n^8/8+n^7/2+(7*n^6)/12-(7*n^4)/24+n^2/12 n^9/9+n^8/2+(2*n^7)/3-(7*n^5)/15+(2*n^3)/9-n/30 n^10/10+n^9/2+(3*n^8)/4-(7*n^6)/10+n^4/2-(3*n^2)/20  Modula-2 Translation of: C# MODULE Faulhaber;FROM EXCEPTIONS IMPORT AllocateSource,ExceptionSource,GetMessage,RAISE;FROM FormatString IMPORT FormatString;FROM Terminal IMPORT WriteString,WriteLn,ReadChar; VAR TextWinExSrc : ExceptionSource; (* Helper Functions *)PROCEDURE Abs(n : INTEGER) : INTEGER;BEGIN IF n < 0 THEN RETURN -n END; RETURN nEND Abs; PROCEDURE Binomial(n,k : INTEGER) : INTEGER;VAR i,num,denom : INTEGER;BEGIN IF (n < 0) OR (k < 0) OR (n < k) THEN RAISE(TextWinExSrc, 0, "Argument Exception.") END; IF (n = 0) OR (k = 0) THEN RETURN 1 END; num := 1; FOR i:=k+1 TO n DO num := num * i END; denom := 1; FOR i:=2 TO n - k DO denom := denom * i END; RETURN num / denomEND Binomial; PROCEDURE GCD(a,b : INTEGER) : INTEGER;BEGIN IF b = 0 THEN RETURN a END; RETURN GCD(b, a MOD b)END GCD; PROCEDURE WriteInteger(n : INTEGER);VAR buf : ARRAY[0..15] OF CHAR;BEGIN FormatString("%i", buf, n); WriteString(buf)END WriteInteger; (* Fraction Handling *)TYPE Frac = RECORD num,denom : INTEGER;END; PROCEDURE InitFrac(n,d : INTEGER) : Frac;VAR nn,dd,g : INTEGER;BEGIN IF d = 0 THEN RAISE(TextWinExSrc, 0, "The denominator must not be zero.") END; IF n = 0 THEN d := 1 ELSIF d < 0 THEN n := -n; d := -d END; g := Abs(GCD(n, d)); IF g > 1 THEN n := n / g; d := d / g END; RETURN Frac{n, d}END InitFrac; PROCEDURE EqualFrac(a,b : Frac) : BOOLEAN;BEGIN RETURN (a.num = b.num) AND (a.denom = b.denom)END EqualFrac; PROCEDURE LessFrac(a,b : Frac) : BOOLEAN;BEGIN RETURN a.num * b.denom < b.num * a.denomEND LessFrac; PROCEDURE NegateFrac(f : Frac) : Frac;BEGIN RETURN Frac{-f.num, f.denom}END NegateFrac; PROCEDURE SubFrac(lhs,rhs : Frac) : Frac;BEGIN RETURN InitFrac(lhs.num * rhs.denom - lhs.denom * rhs.num, rhs.denom * lhs.denom)END SubFrac; PROCEDURE MultFrac(lhs,rhs : Frac) : Frac;BEGIN RETURN InitFrac(lhs.num * rhs.num, lhs.denom * rhs.denom)END MultFrac; PROCEDURE Bernoulli(n : INTEGER) : Frac;VAR a : ARRAY[0..15] OF Frac; i,j,m : INTEGER;BEGIN IF n < 0 THEN RAISE(TextWinExSrc, 0, "n may not be negative or zero.") END; FOR m:=0 TO n DO a[m] := Frac{1, m + 1}; FOR j:=m TO 1 BY -1 DO a[j-1] := MultFrac(SubFrac(a[j-1], a[j]), Frac{j, 1}) END END; IF n # 1 THEN RETURN a[0] END; RETURN NegateFrac(a[0])END Bernoulli; PROCEDURE WriteFrac(f : Frac);BEGIN WriteInteger(f.num); IF f.denom # 1 THEN WriteString("/"); WriteInteger(f.denom) ENDEND WriteFrac; (* Target *)PROCEDURE Faulhaber(p : INTEGER);VAR j,pwr,sign : INTEGER; q,coeff : Frac;BEGIN WriteInteger(p); WriteString(" : "); q := InitFrac(1, p + 1); sign := -1; FOR j:=0 TO p DO sign := -1 * sign; coeff := MultFrac(MultFrac(MultFrac(q, Frac{sign, 1}), Frac{Binomial(p + 1, j), 1}), Bernoulli(j)); IF EqualFrac(coeff, Frac{0, 1}) THEN CONTINUE END; IF j = 0 THEN IF NOT EqualFrac(coeff, Frac{1, 1}) THEN IF EqualFrac(coeff, Frac{-1, 1}) THEN WriteString("-") ELSE WriteFrac(coeff) END END ELSE IF EqualFrac(coeff, Frac{1, 1}) THEN WriteString(" + ") ELSIF EqualFrac(coeff, Frac{-1, 1}) THEN WriteString(" - ") ELSIF LessFrac(Frac{0, 1}, coeff) THEN WriteString(" + "); WriteFrac(coeff) ELSE WriteString(" - "); WriteFrac(NegateFrac(coeff)) END END; pwr := p + 1 - j; IF pwr > 1 THEN WriteString("n^"); WriteInteger(pwr) ELSE WriteString("n") END END; WriteLnEND Faulhaber; (* Main *)VAR i : INTEGER;BEGIN FOR i:=0 TO 9 DO Faulhaber(i) END; ReadCharEND Faulhaber. Output: 0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2 PARI/GP PARI/GP has 2 built in functions: bernfrac(n) for Bernoulli numbers and bernpol(n) for Bernoulli polynomials. Using Bernoulli polynomials in PARI/GP is more simple, clear and much faster. (See version #2). Works with: PARI/GP version 2.9.1 and above Version #1. Using Bernoulli numbers. This version is using "Faulhaber's" formula based on Bernoulli numbers. It's not worth using Bernoulli numbers in PARI/GP, because too much cleaning if you are avoiding "dirty" (but correct) result. Note: Find ssubstr() function here on RC.  \\ Using "Faulhaber's" formula based on Bernoulli numbers. aev 2/7/17\\ In str string replace all occurrences of the search string ssrch with the replacement string srepl. aev 3/8/16sreplace(str,ssrch,srepl)={ my(sn=#str,ssn=#ssrch,srn=#srepl,sres="",Vi,Vs,Vz,vin,vin1,vi,L=List(),tok,zi,js=1); if(sn==0,return("")); if(ssn==0||ssn>sn,return(str)); \\ Vi - found ssrch indexes Vi=sfindalls(str,ssrch); vin=#Vi; if(vin==0,return(str)); vin1=vin+1; Vi=Vec(Vi,vin1); Vi[vin1]=sn+1; for(i=1,vin1, vi=Vi[i]; for(j=js,sn, \\print("ij:",i,"/",j,": ",sres); if(j!=vi, sres=concat(sres,ssubstr(str,j,1)), sres=concat(sres,srepl); js=j+ssn; break) ); \\fend j ); \\fend i return(sres);}B(n)=(bernfrac(n));Comb(n,k)={my(r=0); if(k<=n, r=n!/(n-k)!/k!); return(r)};Faulhaber2(p)={ my(s="",s1="",s2="",c1=0,c2=0); for(j=0,p, c1=(-1)^j*Comb(p+1,j)*B(j); c2=(p+1-j); s2="*n"; if(c1==0, next); if(c2==1, s1="", s1=Str("^",c2)); s=Str(s,c1,s2,s1,"+") ); s=ssubstr(s,1,#s-1); s=sreplace(s,"1*n","n"); s=sreplace(s,"+-","-"); if(p==0, s="n", s=Str("(",s,")/",p+1)); print(p,": ",s);}{\\ Testing:for(i=0,9, Faulhaber2(i))}  Output: 0: n 1: (n^2+n)/2 2: (n^3+3/2*n^2+1/2*n)/3 3: (n^4+2*n^3+n^2)/4 4: (n^5+5/2*n^4+5/3*n^3-1/6*n)/5 5: (n^6+3*n^5+5/2*n^4-1/2*n^2)/6 6: (n^7+7/2*n^6+7/2*n^5-7/6*n^3+1/6*n)/7 7: (n^8+4*n^7+14/3*n^6-7/3*n^4+2/3*n^2)/8 8: (n^9+9/2*n^8+6*n^7-21/5*n^5+2*n^3-3/10*n)/9 9: (n^10+5*n^9+15/2*n^8-7*n^6+5*n^4-3/2*n^2)/10 time = 16 ms.  Version #2. Using Bernoulli polynomials. This version is using the sums of pth powers formula from Bernoulli polynomials. It has small, simple and clear code, and produces instant result.  \\ Using a formula based on Bernoulli polynomials. aev 2/5/17Faulhaber1(m)={ my(B,B1,B2,Bn); Bn=bernpol(m+1); x=n+1; B1=eval(Bn); x=0; B2=eval(Bn); Bn=(B1-B2)/(m+1); if(m==0, Bn=Bn-1); print(m,": ",Bn);}{\\ Testing: for(i=0,9, Faulhaber1(i))}  Output: 0: n 1: 1/2*n^2 + 1/2*n 2: 1/3*n^3 + 1/2*n^2 + 1/6*n 3: 1/4*n^4 + 1/2*n^3 + 1/4*n^2 4: 1/5*n^5 + 1/2*n^4 + 1/3*n^3 - 1/30*n 5: 1/6*n^6 + 1/2*n^5 + 5/12*n^4 - 1/12*n^2 6: 1/7*n^7 + 1/2*n^6 + 1/2*n^5 - 1/6*n^3 + 1/42*n 7: 1/8*n^8 + 1/2*n^7 + 7/12*n^6 - 7/24*n^4 + 1/12*n^2 8: 1/9*n^9 + 1/2*n^8 + 2/3*n^7 - 7/15*n^5 + 2/9*n^3 - 1/30*n 9: 1/10*n^10 + 1/2*n^9 + 3/4*n^8 - 7/10*n^6 + 1/2*n^4 - 3/20*n^2 > ## *** last result computed in 0 ms  Perl use 5.014;use Math::Algebra::Symbols; sub bernoulli_number { my (n) = @_; return 0 if n > 1 && n % 2; my @A; for my m (0 .. n) { A[m] = symbols(1) / (m + 1); for (my j = m ; j > 0 ; j--) { A[j - 1] = j * (A[j - 1] - A[j]); } } return A[0];} sub binomial { my (n, k) = @_; return 1 if k == 0 || n == k; binomial(n - 1, k - 1) + binomial(n - 1, k);} sub faulhaber_s_formula { my (p) = @_; my formula = 0; for my j (0 .. p) { formula += binomial(p + 1, j) * bernoulli_number(j) * symbols('n')**(p + 1 - j); } (symbols(1) / (p + 1) * formula) =~ s/\n/n/gr =~ s/\*\*/^/gr =~ s/\*/ /gr;} foreach my i (0 .. 9) { say "i: ", faulhaber_s_formula(i);} Output: 0: n 1: 1/2 n+1/2 n^2 2: 1/6 n+1/2 n^2+1/3 n^3 3: 1/4 n^2+1/2 n^3+1/4 n^4 4: -1/30 n+1/3 n^3+1/2 n^4+1/5 n^5 5: -1/12 n^2+5/12 n^4+1/2 n^5+1/6 n^6 6: 1/42 n-1/6 n^3+1/2 n^5+1/2 n^6+1/7 n^7 7: 1/12 n^2-7/24 n^4+7/12 n^6+1/2 n^7+1/8 n^8 8: -1/30 n+2/9 n^3-7/15 n^5+2/3 n^7+1/2 n^8+1/9 n^9 9: -3/20 n^2+1/2 n^4-7/10 n^6+3/4 n^8+1/2 n^9+1/10 n^10  Perl 6 Works with: Rakudo version 2018.04.01 sub bernoulli_number(n) { return 1/2 if n == 1; return 0/1 if n % 2; my @A; for 0..n -> m { @A[m] = 1 / (m + 1); for m, m-1 ... 1 -> j { @A[j - 1] = j * (@A[j - 1] - @A[j]); } } return @A[0];} sub binomial(n, k) { k == 0 || n == k ?? 1 !! binomial(n-1, k-1) + binomial(n-1, k);} sub faulhaber_s_formula(p) { my @formula = gather for 0..p -> j { take '(' ~ join('/', (binomial(p+1, j) * bernoulli_number(j)).Rat.nude) ~ ")*n^{p+1 - j}"; } my formula = join(' + ', @formula.grep({!m{'(0/1)*'}})); formula .= subst(rx{ '(1/1)*' }, '', :g); formula .= subst(rx{ '^1'ยป }, '', :g); "1/{p+1} * (formula)";} for 0..9 -> p { say "f(p) = ", faulhaber_s_formula(p);} Output: f(0) = 1/1 * (n) f(1) = 1/2 * (n^2 + n) f(2) = 1/3 * (n^3 + (3/2)*n^2 + (1/2)*n) f(3) = 1/4 * (n^4 + (2/1)*n^3 + n^2) f(4) = 1/5 * (n^5 + (5/2)*n^4 + (5/3)*n^3 + (-1/6)*n) f(5) = 1/6 * (n^6 + (3/1)*n^5 + (5/2)*n^4 + (-1/2)*n^2) f(6) = 1/7 * (n^7 + (7/2)*n^6 + (7/2)*n^5 + (-7/6)*n^3 + (1/6)*n) f(7) = 1/8 * (n^8 + (4/1)*n^7 + (14/3)*n^6 + (-7/3)*n^4 + (2/3)*n^2) f(8) = 1/9 * (n^9 + (9/2)*n^8 + (6/1)*n^7 + (-21/5)*n^5 + (2/1)*n^3 + (-3/10)*n) f(9) = 1/10 * (n^10 + (5/1)*n^9 + (15/2)*n^8 + (-7/1)*n^6 + (5/1)*n^4 + (-3/2)*n^2)  Phix Translation of: C# include builtins\pfrac.e -- (0.8.0+) function bernoulli(integer n) sequence a = {} for m=0 to n do a = append(a,{1,m+1}) for j=m to 1 by -1 do a[j] = frac_mul({j,1},frac_sub(a[j+1],a[j])) end for end for if n!=1 then return a[1] end if return frac_uminus(a[1])end function function binomial(integer n, k) if n<0 or k<0 or n<k then ?9/0 end if if n=0 or k=0 then return 1 end if integer num = 1, denom = 1 for i=k+1 to n do num *= i end for for i=2 to n-k do denom *= i end for return num / denomend function procedure faulhaber(integer p) string res = sprintf("%d : ", p) frac q = {1, p+1} for j=0 to p do frac bj = bernoulli(j) if frac_ne(bj,frac_zero) then frac coeff = frac_mul({binomial(p+1,j),p+1},bj) string s = frac_sprint(coeff) if j=0 then if s="1" then s = "" end if else if s[1]='-' then s[1..1] = " - " else s[1..0] = " + " end if end if res &= s&"n" integer pwr = p+1-j if pwr>1 then res &= sprintf("^%d", pwr) end if end if end for printf(1,"%s\n",{res})end procedure for i=0 to 9 do faulhaber(i)end for Output: 0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2  Python The following implementation does not use Bernoulli numbers, but Stirling numbers of the second kind, based on the relation: ${\displaystyle m^{n}=\sum _{k=0}^{n}S_{n}^{k}(m)_{k}=\sum _{k=0}^{n}S_{n}^{k}k!{m \choose k}}$. Then summing: ${\displaystyle \sum _{j=0}^{m}j^{n}=\sum _{j=0}^{m}\sum _{k=0}^{n}S_{n}^{k}k!{j \choose k}=\sum _{k=0}^{n}S_{n}^{k}k!{m+1 \choose k+1}=\sum _{k=0}^{n}S_{n}^{k}{\frac {(m+1)_{k+1}}{k+1}}}$. One has then to expand the product ${\displaystyle (m+1)_{k+1}}$ in order to get a polynomial in the variable ${\displaystyle m}$. Also, for the sum of ${\displaystyle j^{0}}$, the sum is too large by one (since we start at ${\displaystyle j=0}$), this has to be taken into account. Note: a number of the formulae above are invisible to the majority of browsers, including Chrome, IE/Edge, Safari and Opera. They may (subject to the installation of necessary fronts) be visible to Firefox. from fractions import Fraction def nextu(a): n = len(a) a.append(1) for i in range(n - 1, 0, -1): a[i] = i * a[i] + a[i - 1] return a def nextv(a): n = len(a) - 1 b = [(1 - n) * x for x in a] b.append(1) for i in range(n): b[i + 1] += a[i] return b def sumpol(n): u = [0, 1] v = [[1], [1, 1]] yield [Fraction(0), Fraction(1)] for i in range(1, n): v.append(nextv(v[-1])) t = [0] * (i + 2) p = 1 for j, r in enumerate(u): r = Fraction(r, j + 1) for k, s in enumerate(v[j + 1]): t[k] += r * s yield t u = nextu(u) def polstr(a): s = "" q = False n = len(a) - 1 for i, x in enumerate(reversed(a)): i = n - i if i < 2: m = "n" if i == 1 else "" else: m = "n^%d" % i c = str(abs(x)) if i > 0: if c == "1": c = "" else: m = " " + m if x != 0: if q: t = " + " if x > 0 else " - " s += "%s%s%s" % (t, c, m) else: t = "" if x > 0 else "-" s = "%s%s%s" % (t, c, m) q = True if q: return s else: return "0" for i, p in enumerate(sumpol(10)): print(i, ":", polstr(p)) Output: 0 : n 1 : 1/2 n^2 + 1/2 n 2 : 1/3 n^3 + 1/2 n^2 + 1/6 n 3 : 1/4 n^4 + 1/2 n^3 + 1/4 n^2 4 : 1/5 n^5 + 1/2 n^4 + 1/3 n^3 - 1/30 n 5 : 1/6 n^6 + 1/2 n^5 + 5/12 n^4 - 1/12 n^2 6 : 1/7 n^7 + 1/2 n^6 + 1/2 n^5 - 1/6 n^3 + 1/42 n 7 : 1/8 n^8 + 1/2 n^7 + 7/12 n^6 - 7/24 n^4 + 1/12 n^2 8 : 1/9 n^9 + 1/2 n^8 + 2/3 n^7 - 7/15 n^5 + 2/9 n^3 - 1/30 n 9 : 1/10 n^10 + 1/2 n^9 + 3/4 n^8 - 7/10 n^6 + 1/2 n^4 - 3/20 n^2 Racket Racket will simplify rational numbers; if this code simplifies the expressions too much for your tastes (e.g. you like 1/1 * (n)) then tweak the simplify... clauses to taste. #lang racket/base (require racket/match racket/string math/number-theory) (define simplify-arithmetic-expression (letrec ((s-a-e (match-lambda [(list (and op '+) l ... (list '+ m ...) r ...) (s-a-e (,op ,@l ,@m ,@r))] [(list (and op '+) l ... (? number? n1) m ... (? number? n2) r ...) (s-a-e (,op ,@l ,(+ n1 n2) ,@m ,@r))] [(list (and op '+) (app s-a-e l _) ... 0 (app s-a-e r _) ...) (s-a-e (,op ,@l ,@r))] [(list (and op '+) (app s-a-e x _)) (values x #t)] [(list (and op '*) l ... (list '* m ...) r ...) (s-a-e (,op ,@l ,@m ,@r))] [(list (and op '*) l ... (? number? n1) m ... (? number? n2) r ...) (s-a-e (,op ,@l ,(* n1 n2) ,@m ,@r))] [(list (and op '*) (app s-a-e l _) ... 1 (app s-a-e r _) ...) (s-a-e (,op ,@l ,@r))] [(list (and op '*) (app s-a-e l _) ... 0 (app s-a-e r _) ...) (values 0 #t)] [(list (and op '*) (app s-a-e x _)) (values x #t)] [(list 'expt (app s-a-e x x-simplified?) 1) (values x x-simplified?)] [(list op (app s-a-e a #f) ...) (values (,op ,@a) #f)] [(list op (app s-a-e a _) ...) (s-a-e (,op ,@a))] [e (values e #f)]))) s-a-e)) (define (expression->infix-string e) (define (parenthesise-maybe s p?) (if p? (string-append "(" s ")") s)) (letrec ((e->is (lambda (paren?) (match-lambda [(list (and op (or '+ '- '* '*)) (app (e->is #t) a p?) ...) (define bits (map parenthesise-maybe a p?)) (define compound (string-join bits (format " ~a " op))) (values (if paren? (string-append "(" compound ")") compound) #f)] [(list 'expt (app (e->is #t) x xp?) (app (e->is #t) n np?)) (values (format "~a^~a" (parenthesise-maybe x xp?) (parenthesise-maybe n np?)) #f)] [(? number? (app number->string s)) (values s #f)] [(? symbol? (app symbol->string s)) (values s #f)])))) (define-values (str needs-parens?) ((e->is #f) e)) str)) (define (faulhaber p) (define p+1 (add1 p)) (define-values (simpler simplified?) (simplify-arithmetic-expression (* ,(/ 1 p+1) (+ ,@(for/list ((j (in-range p+1))) (* ,(* (expt -1 j) (binomial p+1 j)) (* ,(bernoulli-number j) (expt n ,(- p+1 j))))))))) simpler) (for ((p (in-range 0 (add1 9)))) (printf "f(~a) = ~a~%" p (expression->infix-string (faulhaber p))))  Output: f(0) = n f(1) = 1/2 * (n^2 + n) f(2) = 1/3 * (n^3 + (3/2 * n^2) + (1/2 * n)) f(3) = 1/4 * (n^4 + (2 * n^3) + n^2) f(4) = 1/5 * (n^5 + (5/2 * n^4) + (5/3 * n^3) + (-1/6 * n)) f(5) = 1/6 * (n^6 + (3 * n^5) + (5/2 * n^4) + (-1/2 * n^2)) f(6) = 1/7 * (n^7 + (7/2 * n^6) + (7/2 * n^5) + (-7/6 * n^3) + (1/6 * n)) f(7) = 1/8 * (n^8 + (4 * n^7) + (14/3 * n^6) + (-7/3 * n^4) + (2/3 * n^2)) f(8) = 1/9 * (n^9 + (9/2 * n^8) + (6 * n^7) + (-21/5 * n^5) + (2 * n^3) + (-3/10 * n)) f(9) = 1/10 * (n^10 + (5 * n^9) + (15/2 * n^8) + (-7 * n^6) + (5 * n^4) + (-3/2 * n^2))  Sidef func faulhaber_s_formula(p) { var formula = gather { { |j| take "(#{binomial(p+1, j) * j.bernfrac -> as_rat})*n^#{p+1 - j}" } << 0..p } formula.grep! { !.contains('(0)*') }.join!(' + ') formula -= /\(1$$\*/g    formula -= /\^1\b/g    formula.gsub!(/$$([^+]*?)$$/, { _ })     "1/#{p + 1} * (#{formula})"} { |p|    printf("%2d: %s\n", p, faulhaber_s_formula(p))} << ^10
Output:
 0: 1/1 * (n)
1: 1/2 * (n^2 + n)
2: 1/3 * (n^3 + 3/2*n^2 + 1/2*n)
3: 1/4 * (n^4 + 2*n^3 + n^2)
4: 1/5 * (n^5 + 5/2*n^4 + 5/3*n^3 + -1/6*n)
5: 1/6 * (n^6 + 3*n^5 + 5/2*n^4 + -1/2*n^2)
6: 1/7 * (n^7 + 7/2*n^6 + 7/2*n^5 + -7/6*n^3 + 1/6*n)
7: 1/8 * (n^8 + 4*n^7 + 14/3*n^6 + -7/3*n^4 + 2/3*n^2)
8: 1/9 * (n^9 + 9/2*n^8 + 6*n^7 + -21/5*n^5 + 2*n^3 + -3/10*n)
9: 1/10 * (n^10 + 5*n^9 + 15/2*n^8 + -7*n^6 + 5*n^4 + -3/2*n^2)


By not simplifying the formulas, we can have a much cleaner code:

func faulhaber_s_formula(p) {    "1/#{p + 1} * (" + gather {      { |j|         take "#{binomial(p+1, j) * j.bernfrac -> as_rat}*n^#{p+1 - j}"      } << 0..p    }.join(' + ') + ")"} { |p|    printf("%2d: %s\n", p, faulhaber_s_formula(p))} << ^10
Output:
 0: 1/1 * (1*n^1)
1: 1/2 * (1*n^2 + 1*n^1)
2: 1/3 * (1*n^3 + 3/2*n^2 + 1/2*n^1)
3: 1/4 * (1*n^4 + 2*n^3 + 1*n^2 + 0*n^1)
4: 1/5 * (1*n^5 + 5/2*n^4 + 5/3*n^3 + 0*n^2 + -1/6*n^1)
5: 1/6 * (1*n^6 + 3*n^5 + 5/2*n^4 + 0*n^3 + -1/2*n^2 + 0*n^1)
6: 1/7 * (1*n^7 + 7/2*n^6 + 7/2*n^5 + 0*n^4 + -7/6*n^3 + 0*n^2 + 1/6*n^1)
7: 1/8 * (1*n^8 + 4*n^7 + 14/3*n^6 + 0*n^5 + -7/3*n^4 + 0*n^3 + 2/3*n^2 + 0*n^1)
8: 1/9 * (1*n^9 + 9/2*n^8 + 6*n^7 + 0*n^6 + -21/5*n^5 + 0*n^4 + 2*n^3 + 0*n^2 + -3/10*n^1)
9: 1/10 * (1*n^10 + 5*n^9 + 15/2*n^8 + 0*n^7 + -7*n^6 + 0*n^5 + 5*n^4 + 0*n^3 + -3/2*n^2 + 0*n^1)


zkl

Library: GMP
GNU Multiple Precision Arithmetic Library

Uses code from the Bernoulli numbers task (copied here).

var [const] BN=Import("zklBigNum");	// libGMP (GNU MP Bignum Library) fcn faulhaberFormula(p){  //-->(Rational,Rational...)   [p..0,-1].pump(List(),'wrap(k){ B(k)*BN(p+1).binomial(k) })   .apply('*(Rational(1,p+1)))}
foreach p in (10){    println("F(%d) --> %s".fmt(p,polyRatString(faulhaberFormula(p))))}
class Rational{  // Weenie Rational class, can handle BigInts   fcn init(_a,_b){ var a=_a, b=_b; normalize(); }   fcn toString{       if(b==1) a.toString()      else     "%d/%d".fmt(a,b)    }   var [proxy] isZero=fcn{ a==0   };   fcn normalize{  // divide a and b by gcd      g:= a.gcd(b);      a/=g; b/=g;      if(b<0){ a=-a; b=-b; } // denominator > 0      self   }   fcn __opAdd(n){      if(Rational.isChildOf(n)) self(a*n.b + b*n.a, b*n.b); // Rat + Rat      else self(b*n + a, b);				    // Rat + Int   }   fcn __opSub(n){ self(a*n.b - b*n.a, b*n.b) }		    // Rat - Rat   fcn __opMul(n){      if(Rational.isChildOf(n)) self(a*n.a, b*n.b);	    // Rat * Rat      else self(a*n, b);				    // Rat * Int   }   fcn __opDiv(n){ self(a*n.b,b*n.a) }			    // Rat / Rat}
fcn B(N){	// calculate Bernoulli(n) --> Rational   var A=List.createLong(100,0);  // aka static aka not thread safe   foreach m in (N+1){      A[m]=Rational(BN(1),BN(m+1));      foreach j in ([m..1, -1]){ A[j-1]= (A[j-1] - A[j])*j; }   }   A[0]}fcn polyRatString(terms){ // (a1,a2...)-->"a1n + a2n^2 ..."   str:=[1..].zipWith('wrap(n,a){ if(a.isZero) "" else "+ %sn^%s ".fmt(a,n) },        terms)   .pump(String)   .replace(" 1n"," n").replace("n^1 ","n ").replace("+ -","- ");   if(not str)     return(" ");  // all zeros   if(str[0]=="+") str[1,*];     // leave leading space   else            String("-",str[2,*]);}
Output:
F(0) -->  n
F(1) -->  1/2n + 1/2n^2
F(2) -->  1/6n + 1/2n^2 + 1/3n^3
F(3) -->  1/4n^2 + 1/2n^3 + 1/4n^4
F(4) --> -1/30n + 1/3n^3 + 1/2n^4 + 1/5n^5
F(5) --> -1/12n^2 + 5/12n^4 + 1/2n^5 + 1/6n^6
F(6) -->  1/42n - 1/6n^3 + 1/2n^5 + 1/2n^6 + 1/7n^7
F(7) -->  1/12n^2 - 7/24n^4 + 7/12n^6 + 1/2n^7 + 1/8n^8
F(8) --> -1/30n + 2/9n^3 - 7/15n^5 + 2/3n^7 + 1/2n^8 + 1/9n^9
F(9) --> -3/20n^2 + 1/2n^4 - 7/10n^6 + 3/4n^8 + 1/2n^9 + 1/10n^10